A371647 -id:A371647 - cp's OEIS frontend

A371648 Decimal expansion of Product_{k>=0} (1 + 1/Fibonacci(5^k)).

2, 4, 0, 0, 0, 3, 1, 9, 8, 9, 3, 3, 6, 8, 8, 7, 7, 0, 4, 0, 9, 8, 6, 3, 3, 8, 2, 9, 1, 2, 4, 5, 9, 0, 4, 4, 8, 8, 5, 5, 4, 9, 7, 8, 3, 1, 9, 3, 3, 8, 7, 6, 7, 8, 8, 4, 2, 5, 9, 6, 1, 1, 5, 6, 8, 7, 9, 3, 5, 0, 3, 7, 9, 0, 2, 9, 3, 0, 1, 3, 9, 6, 1, 0, 0, 0, 6, 4, 3, 0, 2, 5, 1, 3, 1, 8, 3, 6, 0, 7, 8, 0, 0, 4, 1

Offset: 1

Amiram Eldar, Mar 31 2024

This constant is a transcendental number (Nyblom, 2004).

			2.40003198933688770409863382912459044885549783193387...

M. A. Nyblom, On the Construction of a Family of Transcendental Valued Infinite Products, Fibonacci Quarterly, Vol. 42, No. 4 (2004), pp. 353-358.
Index entries for transcendental numbers.

Cf. A000045, A145232, A371647.

Similar constants: A337668, A337669, A371650.

RealDigits[Product[1 + 1/Fibonacci[5^k], {k, 0, 10}], 10, 120][[1]]

PARI
```
prodinf(k = 0, 1 + 1/fibonacci(5^k))
```

Equals Product_{k>=0} (1 + 1/A145232(k)).

A371649 Decimal expansion of Sum_{k>=0} 1/Lucas(5^k).

Original entry on oeis.org

1, 0, 9, 0, 9, 1, 5, 0, 5, 1, 7, 7, 0, 0, 7, 7, 6, 7, 0, 0, 1, 8, 6, 5, 7, 5, 0, 2, 4, 1, 4, 2, 2, 8, 2, 0, 5, 7, 1, 5, 1, 7, 5, 1, 0, 2, 3, 1, 9, 9, 0, 6, 6, 9, 8, 9, 0, 5, 0, 3, 2, 1, 7, 0, 9, 2, 2, 2, 4, 3, 0, 8, 1, 7, 5, 8, 2, 8, 8, 4, 4, 6, 4, 9, 0, 2, 6, 3, 1, 9, 6, 7, 3, 7, 2, 4, 8, 1, 8, 3, 1, 2, 4, 1, 7

Offset: 1

Amiram Eldar, Mar 31 2024

This constant is a transcendental number (Nyblom, 2001).

			1.09091505177007767001865750241422820571517510231990...

M. A. Nyblom, A Theorem on Transcendence of Infinite Series II, Journal of Number Theory, Vol. 91, No. 1 (2001), pp. 71-80.
Index entries for transcendental numbers.

Cf. A000032, A144837, A371650.

Similar constants: A093540, A338304, A338612, A371647.

RealDigits[Sum[1/LucasL[5^k], {k, 0, 10}], 10, 120][[1]]

suminf(k = 0, 1/(fibonacci(5^k-1) + fibonacci(5^k+1)))

Equals Sum_{k>=0} 1/A144837(k).

cp's OEIS Frontend

A371648 Decimal expansion of Product_{k>=0} (1 + 1/Fibonacci(5^k)).

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